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Theorem mon1pval 20017
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p  |-  P  =  (Poly1 `  R )
uc1pval.b  |-  B  =  ( Base `  P
)
uc1pval.z  |-  .0.  =  ( 0g `  P )
uc1pval.d  |-  D  =  ( deg1  `  R )
mon1pval.m  |-  M  =  (Monic1p `  R )
mon1pval.o  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mon1pval  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Distinct variable groups:    B, f    D, f    .1. , f    R, f    .0. , f
Allowed substitution hints:    P( f)    M( f)

Proof of Theorem mon1pval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mon1pval.m . 2  |-  M  =  (Monic1p `  R )
2 fveq2 5687 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2454 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5691 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2454 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5691 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2454 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2581 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5687 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2454 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5689 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5691 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5687 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
18 mon1pval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
1917, 18syl6eqr 2454 . . . . . . 7  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2016, 19eqeq12d 2418 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( 1r `  r
)  <->  ( (coe1 `  f
) `  ( D `  f ) )  =  .1.  ) )
2111, 20anbi12d 692 . . . . 5  |-  ( r  =  R  ->  (
( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) )  <->  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  )
) )
227, 21rabeqbidv 2911 . . . 4  |-  ( r  =  R  ->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) }  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
23 df-mon1 20006 . . . 4  |- Monic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) } )
24 fvex 5701 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2474 . . . . 5  |-  B  e. 
_V
2625rabex 4314 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  e.  _V
2722, 23, 26fvmpt 5765 . . 3  |-  ( R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
28 fvprc 5681 . . . 4  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  (/) )
29 ssrab2 3388 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  B
30 fvprc 5681 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2448 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5691 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
336, 32syl5eq 2448 . . . . . . 7  |-  ( -.  R  e.  _V  ->  B  =  ( Base `  (/) ) )
34 base0 13461 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3533, 34syl6eqr 2454 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3356 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/) )
37 ss0 3618 . . . . 5  |-  ( { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/)  ->  { f  e.  B  |  (
f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3836, 37syl 16 . . . 4  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3928, 38eqtr4d 2439 . . 3  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
4027, 39pm2.61i 158 . 2  |-  (Monic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }
411, 40eqtri 2424 1  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ` cfv 5413   Basecbs 13424   0gc0g 13678   1rcur 15617  Poly1cpl1 16526  coe1cco1 16529   deg1 cdg1 19930  Monic1pcmn1 20001
This theorem is referenced by:  ismon1p  20018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-slot 13428  df-base 13429  df-mon1 20006
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